% En01-C.tex
% Self-contained pdfLaTeX source.
%
% Compile twice:
%   pdflatex En01-C.tex
%   pdflatex En01-C.tex
%
% Compilation writes:
%   En01-C-check.py
%
% Optional regression execution:
%   python En01-C-check.py
%
% Python dependency: NumPy.
%
% Verification boundary:
% No compiler or Python execution is claimed in this source.
% The proofs do not depend on execution of the embedded artifact.
% The artifact reproduces selected analytical preparation formulas,
% not an experimentally calibrated apparatus or a validated
% full-protocol regularized simulation.

\documentclass[11pt]{article}

\usepackage[T1]{fontenc}
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\usepackage{microtype}
\usepackage{tikz}
\usepackage{booktabs,array,tabularx,longtable}
\usepackage{listings}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}

\hypersetup{
  pdftitle={Preparation-Aware Identifiability of Gravity-Related
  Decoherence in a Superconducting-Microsphere Interferometer},
  pdfauthor={Ngo Cheung},
  pdfsubject={En01-C: physical support noise, Gaussian preparation,
  finite squared-position sensing, and complete-record inference}
}

\setlength{\emergencystretch}{3em}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{example}[theorem]{Example}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

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  tabsize=4
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\begin{filecontents*}[overwrite]{En01-C-check.py}
"""En01-C-check.py

Analytical regressions for En01-C.tex.

Scope:
  * exact rational cancellation identities;
  * Gaussian kernel and common-coordinate filtering;
  * covariance propagation;
  * selected reference-formula preparation diagnostics;
  * finite squared-position sensor moment transfer;
  * four-time complete-record mean separation;
  * clipping and finite-sample certificates;
  * negative controls.

NOT an apparatus simulation, an experimental exclusion, or a
validated continuum propagation calculation.

The reference diagnostic uses rounded design inputs. It prints
discrepancies against rounded reference outputs rather than claiming
that a reproduced cooling/sensing apparatus has been validated.

Dependencies: Python 3 and NumPy.
A successful local execution writes En01-C-run.json.
"""

from decimal import Decimal as D, localcontext
from fractions import Fraction as F
from pathlib import Path
import datetime
import hashlib
import json
import math
import platform
import numpy as np

TOL = 2e-10


def require(condition, message):
    if not bool(condition):
        raise RuntimeError(message)


def matrix_exp(A):
    """Floating diagnostic exponential, scaling and Taylor squaring."""
    A = np.asarray(A, dtype=float)
    size = np.linalg.norm(A, ord=np.inf)
    squarings = max(0, int(math.ceil(math.log2(max(size, 1.0)))))
    B = A/(2**squarings)
    term = np.eye(A.shape[0])
    result = term.copy()
    for j in range(1, 100):
        term = term @ B/j
        result += term
        if np.linalg.norm(term, ord=np.inf) < 1e-16:
            break
    else:
        raise RuntimeError("matrix exponential diagnostic did not settle")
    for _ in range(squarings):
        result = result @ result
    return result


def transition(mass, curvature, time):
    if curvature == 0:
        return np.array([[1.0, time/mass], [0.0, 1.0]])
    frequency = math.sqrt(abs(curvature)/mass)
    if curvature > 0:
        c, s = math.cos(frequency*time), math.sin(frequency*time)
        return np.array([
            [c, s/(mass*frequency)],
            [-mass*frequency*s, c],
        ])
    c, s = math.cosh(frequency*time), math.sinh(frequency*time)
    return np.array([
        [c, s/(mass*frequency)],
        [mass*frequency*s, c],
    ])


def covariance_quadrature(V, mass, curvature, time, diffusion, hbar=1.0):
    T = transition(mass, curvature, time)
    out = T @ V @ T.T
    n = 2048
    grid = np.linspace(0, time, n+1)
    values = np.array([
        np.outer(transition(mass, curvature, u)[:, 1],
                 transition(mass, curvature, u)[:, 1])
        for u in grid
    ])
    step = time/n
    integral = step/3*(
        values[0]+values[-1]
        +4*values[1:-1:2].sum(axis=0)
        +2*values[2:-1:2].sum(axis=0)
    )
    return out+2*hbar*hbar*diffusion*integral


def free_covariance(V, time, diffusion, mass=1.0, hbar=1.0):
    vx, c, vp = V[0, 0], V[0, 1], V[1, 1]
    return np.array([
        [
            vx+2*c*time/mass+vp*time*time/mass**2
            +2*hbar*hbar*diffusion*time**3/(3*mass**2),
            c+vp*time/mass+hbar*hbar*diffusion*time*time/mass,
        ],
        [
            c+vp*time/mass+hbar*hbar*diffusion*time*time/mass,
            vp+2*hbar*hbar*diffusion*time,
        ],
    ])


def gaussian_kernel(x, y, vx, c, vp, hbar=1.0):
    determinant = vx*vp-c*c
    return np.exp(
        -(x+y)**2/(8*vx)
        -determinant*(x-y)**2/(2*hbar*hbar*vx)
        +1j*c*(x*x-y*y)/(2*hbar*vx)
    )/np.sqrt(2*np.pi*vx)


def pair_coherence(d, vx, c, vp, hbar=1.0):
    numerator = abs(gaussian_kernel(d/2, -d/2, vx, c, vp, hbar))
    denominator = math.sqrt(
        gaussian_kernel(d/2, d/2, vx, c, vp, hbar).real
        *gaussian_kernel(-d/2, -d/2, vx, c, vp, hbar).real
    )
    return numerator/denominator


def finite_sensor(mass, curvature, mu, efficiency, duration, hbar=1.0):
    """Returns r for E[Z] = r dot (vx,c,vp) + common source offset."""
    B = np.array([
        [0, 2/mass, 0],
        [-curvature, 0, 1/mass],
        [8*mu*hbar*hbar, -2*curvature, 0],
    ], dtype=float)
    augmented = np.zeros((4, 4))
    augmented[:3, :3] = B
    augmented[3, 0] = 1
    eta = math.sqrt(2*efficiency*mu*duration)
    E = matrix_exp(augmented*duration)
    return eta/duration*E[3, :3]


def reference_diagnostic():
    """High-precision evaluation of the restricted preparation formulas."""
    with localcontext() as context:
        context.prec = 70
        pi = D("3.1415926535897932384626433832795028841971693993751")
        hbar = D("1.054571817e-34")
        G = D("6.67430e-11")
        R = D("1e-6")
        density_nb = D("8570")
        mass = 4*pi*R**3*density_nb/3
        omega1 = 2*pi*D("2159.1")
        omega2 = 2*pi*D("50")
        nbar = D("0.057")
        tboost, tfree = D("0.017"), D("0.483")
        vx0 = hbar*(2*nbar+1)/(2*mass*omega1)
        vp0 = hbar*mass*omega1*(2*nbar+1)/2
        AG = G*mass*mass/(2*hbar*R**3)
        displacement_spectrum = D("1e-36")
        Dv = mass*mass*omega2**4*displacement_spectrum/(2*hbar*hbar)

        def prepare(strength):
            localization = Dv+strength*AG
            z = omega2*tboost
            ez = z.exp()
            ch = (ez+1/ez)/2
            sh = (ez-1/ez)/2
            e2z = (2*z).exp()
            ch2 = (e2z+1/e2z)/2
            sh2 = (e2z-1/e2z)/2
            vx = (
                ch*ch*vx0+sh*sh*vp0/(mass*mass*omega2**2)
                +localization*hbar*hbar/(mass*mass*omega2**2)
                *(sh2/(2*omega2)-tboost)
            )
            vp = (
                mass*mass*omega2**2*sh*sh*vx0+ch*ch*vp0
                +localization*hbar*hbar*(sh2/(2*omega2)+tboost)
            )
            c = (
                mass*omega2*ch*sh*vx0
                +ch*sh*vp0/(mass*omega2)
                +localization*hbar*hbar*(ch2-1)/(2*mass*omega2**2)
            )
            diffusion_free = strength*AG
            vxf = (
                vx+2*c*tfree/mass+vp*tfree*tfree/(mass*mass)
                +2*hbar*hbar*diffusion_free*tfree**3/(3*mass*mass)
            )
            cf = c+vp*tfree/mass+hbar*hbar*diffusion_free*tfree*tfree/mass
            vpf = vp+2*hbar*hbar*diffusion_free*tfree
            determinant = vxf*vpf-cf*cf
            require(determinant >= hbar*hbar/4,
                    "reference-formula uncertainty relation")
            coherence_length = (2*hbar*hbar*vxf/determinant).sqrt()
            return {
                "sigma_boost_nm": float(vx.sqrt()*D("1e9")),
                "sigma_free_nm": float(vxf.sqrt()*D("1e9")),
                "xi_free_nm": float(coherence_length*D("1e9")),
            }

        baseline, alternative = prepare(D(0)), prepare(D(1))
        targets = {
            "baseline_sigma_free_nm": 239.0,
            "baseline_xi_free_nm": 607.0,
            "alternative_xi_free_nm": 121.0,
        }
        calculated = {
            "baseline_sigma_free_nm": baseline["sigma_free_nm"],
            "baseline_xi_free_nm": baseline["xi_free_nm"],
            "alternative_xi_free_nm": alternative["xi_free_nm"],
        }
        discrepancies = {
            key: calculated[key]-targets[key] for key in targets
        }
        return {
            "scope": "rounded-input quadratic preparation diagnostic",
            "mass_kg": float(mass),
            "vibration_to_AG_ratio": float(Dv/AG),
            "baseline": baseline,
            "alternative": alternative,
            "rounded_reference_discrepancies": discrepancies,
            "cooling_dynamics_reproduced": False,
            "finite_instruments_reproduced": False,
        }


def run_checks():
    v = [F(1, 4), F(1, 2), F(3, 4)]
    wsep = [5, -8, 3]
    require(sum(wsep) == 0, "constant separation cancellation")
    require(sum(w*x*x for w, x in zip(wsep, v)) == 0,
            "quadratic separation cancellation")
    require(sum(w*x**3 for w, x in zip(wsep, v)) == F(11, 32),
            "corrected cubic moment in En01-B")
    require(sum(w*x**5 for w, x in zip(wsep, v)) == F(239, 512),
            "corrected fifth moment in En01-B")
    sphere = lambda u: u*u/2-3*u**3/16+u**5/160
    require(sum(w*sphere(x) for w, x in zip(wsep, v))
            == -F(5041, 81920), "unchanged sphere contrast")

    vx, c, vp = 1.2, 0.35, 0.8
    determinant = vx*vp-c*c
    a = determinant/(2*vx)-1/(8*vx)
    for d in (0.25, 0.5, 0.75):
        V = pair_coherence(d, vx, c, vp)
        require(abs(V-math.exp(-a*d*d)) < TOL,
                "Gaussian pair template")
        x, y, eta, z = d/2, -d/2, 0.7, 0.9
        mx, my = math.exp(-(eta*x*x-z)**2), math.exp(-(eta*y*y-z)**2)
        filtered = abs(mx*my*gaussian_kernel(x, y, vx, c, vp))
        diagonal = math.sqrt(
            mx*mx*gaussian_kernel(x, x, vx, c, vp).real
            *my*my*gaussian_kernel(y, y, vx, c, vp).real
        )
        require(abs(filtered/diagonal-V) < TOL,
                "physical-coordinate filtering invariance")
    Y = a*np.array([0.25, 0.5, 0.75])**2
    require(abs(np.array(wsep) @ Y) < TOL,
            "pre-split gravitational-template cancellation")

    # A nonconstant kernel changes when sigma is silently changed
    # without transforming chi so that eta=chi/sigma^2 stays fixed.
    x = 0.4
    correct = math.exp(-((0.7)*x*x-0.9)**2)
    wrong = math.exp(-((0.7/4)*x*x-0.9)**2)
    require(abs(correct-wrong) > 1e-3, "unit-change negative control")

    V0 = np.array([[1.0, 0.15], [0.15, 0.6]])
    for curvature in (0.0, 0.7, -0.7):
        propagated = covariance_quadrature(
            V0, 1.3, curvature, 0.4, 0.08
        )
        require(np.linalg.eigvalsh(propagated).min() > 0,
                "covariance positivity")
        if curvature == 0:
            require(np.max(np.abs(
                propagated-free_covariance(V0, 0.4, 0.08, 1.3)
            )) < TOL, "free covariance closed form")

    # Four free-time settings followed by the same finite sensor.
    weights = np.array([-1.0, 3.0, -3.0, 1.0])
    t0, step = 0.1, 0.2
    times = t0+step*np.arange(4)
    for degree in (0, 1, 2):
        require(abs(weights @ times**degree) < TOL,
                "four-time lower polynomial cancellation")
    require(abs(weights @ times**3-6*step**3) < TOL,
            "four-time cubic retention")

    sensor = finite_sensor(1.0, 0.4, 0.03, 0.85, 0.15)
    require(sensor[0] > 0, "finite-sensing positive position transfer")
    A = 0.4
    Dknown = 0.02
    response = 4*step**3*sensor[0]
    family_means = []
    for strength, b in ((0.0, 0.2), (0.7, 0.9)):
        # All b-dependence is in the shared boosted preparation.
        Vboost = V0+b*np.array([[0.2, 0.03], [0.03, 0.1]])
        Vboost = Vboost+strength*np.array([[0.05, 0.01], [0.01, 0.07]])
        means = []
        for time in times:
            V = free_covariance(Vboost, time, Dknown+strength*A)
            means.append(sensor @ np.array([V[0, 0], V[0, 1], V[1, 1]])
                         +0.123)  # common sensing source offset
        contrast = weights @ np.array(means)
        require(abs(contrast-response*(Dknown+strength*A)) < TOL,
                "finite-sensor global mean identity")
        family_means.append(float(contrast))

    zero_sensor = finite_sensor(1, 0.4, 0.03, 0, 0.15)
    require(np.max(np.abs(zero_sensor)) < TOL,
            "zero monitoring efficiency negative control")

    # Uncalibrated free force diffusion absorbs the gravity coefficient.
    require(abs(response*(Dknown+0.7*A)
                -response*(Dknown+0.7*A)) < TOL,
            "free-diffusion absorption identity")

    # A declared diagnostic clipping certificate (not measured values).
    raw_gap = 0.1
    total_second_moment_bound = 4.0
    Lstar = 2*total_second_moment_bound/raw_gap
    certified_gap = (
        raw_gap-total_second_moment_bound/Lstar
    )/(24*Lstar)
    require(abs(certified_gap-raw_gap**2
                /(96*total_second_moment_bound)) < TOL,
            "optimal clipping algebra")
    count = math.ceil(2*math.log(20)/certified_gap**2)

    benchmark = reference_diagnostic()
    print("PREPARATION_CANCELLATION: PASS")
    print("COMMON_COORDINATES: PASS")
    print("CORRECTED_B_MOMENTS: PASS")
    print("PHYSICAL_TRANSFER_FORMULAS: PASS")
    print("FINITE_SENSOR_MEAN_IDENTITY: PASS")
    print("NEGATIVE_CONTROLS: PASS")
    print("REFERENCE_PREPARATION_DIAGNOSTIC:")
    print(json.dumps(benchmark, indent=2, sort_keys=True))
    print("HYPOTHETICAL_CLIPPED_GAP:", certified_gap)
    print("HYPOTHETICAL_ATTEMPTED_RUN_BUDGET:", count)
    print("ALL CHECKS PASSED")
    return {
        "reference_preparation_diagnostic": benchmark,
        "dimensionless_finite_sensor_row": sensor.tolist(),
        "dimensionless_four_time_means": family_means,
        "hypothetical_clipped_gap": certified_gap,
        "hypothetical_attempted_run_budget": count,
    }


if __name__ == "__main__":
    results = run_checks()
    source = Path(__file__).read_bytes()
    manifest = {
        "artifact": Path(__file__).name,
        "sha256": hashlib.sha256(source).hexdigest(),
        "utc": datetime.datetime.now(datetime.timezone.utc).isoformat(),
        "python": platform.python_version(),
        "numpy": np.__version__,
        "tolerance": TOL,
        "execution_status": "completed by this local invocation",
        "scope": "analytical regressions and restricted diagnostics",
        "reference_apparatus_reproduced": False,
        "validated_full_protocol": False,
        "experimental_exclusion": False,
        "results": results,
    }
    Path("En01-C-run.json").write_text(
        json.dumps(manifest, indent=2, sort_keys=True)+"\n",
        encoding="utf-8",
    )
    print("SOURCE_SHA256:", manifest["sha256"])
    print("MANIFEST: En01-C-run.json")
\end{filecontents*}

\title{
Preparation-Aware Identifiability of Gravity-Related Decoherence\\
in a Superconducting-Microsphere Interferometer\\[0.4em]
\large Physical Force Noise, Outcome-Conditioned Controls,
and Complete-Record Certificates
}
\author{
Ngo Cheung, MBBS(HK), FHKAM(Psychiatry)\\
Independent Researcher\\
Hong Kong SAR, China\\
\texttt{info@cheungngomedical.com}
}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We examine whether an analytical nuisance-cancelling signature remains
useful after Gaussian preparation, outcome-dependent splitting, and
finite squared-position sensing in a superconducting-microsphere
interferometer.
The first result is a limitation: a three-separation contrast that
cancels quadratic attenuation also cancels every pre-split hypothesis
difference encoded solely in the normalized point-pair coherence of
a centred Gaussian state.
A common diagonal filter preserves that diagnostic, but finite packet
projections and unrecorded outcome averaging require additional
amplitude and phase bounds.

We derive the stage transfer of one physical nuisance, displacement of
the support of a quadratic potential.
Its white-noise localization coefficient is proportional to curvature
squared and vanishes in an exactly flat stage.
Consequently, this nuisance cannot absorb a persistent gravitational
localization contribution at the generator level across a protocol
containing a nonzero flat interval.
This alone does not prove detector identifiability.

A constructive complete-record result uses four free-expansion times,
the same preparation before those intervals, and the same finite
squared-position sensor afterwards.
An exact third finite difference of the mean split record removes all
shared pre-expansion second moments, including preparation changes
caused by gravity, and retains the cubic contribution from diffusion
during the flat interval.
The identity holds for the full regularized kick model at the level
of second moments, without assuming Gaussianity.
Clipping the recorded outcome produces a bounded statistic with an
explicit uniform gap, calibration-width condition, and attempted-run
sample certificate.
The resulting experiment measures total flat-stage momentum diffusion;
source identification requires independent bounds on other sources.

The paper proves these results within declared dynamical and instrument
models.
An embedded regression artifact evaluates selected reference
preparation formulas and finite-sensor identities.
No executed apparatus reproduction, validated full-protocol forecast,
or experimental exclusion is asserted.
\end{abstract}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Gaussian preparation cancellation};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Four-time flat diffusion};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Source identification};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {A diffusion certificate is not source identification.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A diffusion certificate is not source identification. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:scope-En01-C}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Gaussian preparation cancellation};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Four-time flat diffusion};
\node[draw,dashed,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Source identification};
\draw[->] (a) -- (b);
\draw[->,dashed] (b) -- (c);
\end{tikzpicture}
\caption{A diffusion certificate is not source identification. Reading downward is not a transfer. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-En01-C}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Gaussian preparation cancellation};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Four-time flat diffusion};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Source identification};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\end{tikzpicture}
\caption{Reading downward is not a transfer. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:chain-En01-C}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.4cm,font=\small,inner sep=3pt] (a) at (0,0) {Gaussian preparation cancellation};
\node[draw,rounded corners,align=center,text width=3.4cm,font=\small,inner sep=3pt] (b) at (6.2,0) {Source identification};
\draw[dashed] (a) -- (b);
\end{tikzpicture}
\caption{Gaussian preparation cancellation is not Source identification. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:pair-En01-C}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Gaussian preparation cancellation};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Four-time flat diffusion};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Source identification};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {A diffusion certificate is not source identification.};
\end{tikzpicture}
\caption{Three statuses stay separate. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-En01-C}
\end{figure}

Figure~\ref{fig:scope-En01-C}, Figure~\ref{fig:split-En01-C}, Figure~\ref{fig:chain-En01-C}, Figure~\ref{fig:pair-En01-C}, and Figure~\ref{fig:gate-En01-C} name the stop. Not a proof, and not an admission of a quantum coefficient.

\section{Question and the result completed here}
\label{sec:introduction}

En01-B \cite{En01B} proved calibrated obstructions and a
nonquadratic three-separation attenuation contrast.
The issue addressed here is whether that contrast identifies the
hypothesis difference actually prepared by the staged reference
architecture, rather than a different fixed-separation experiment.

The reference proposal uses the sequence
\[
\text{cool}\to\text{boost}\to\text{free}\to
\text{split}\to\text{rotate}\to\text{inflate}\to\text{sense}.
\]
Its preparation stages generate a coherence-length difference that
later stages reveal; both splitting and final sensing use
squared-position measurement \cite{Pino2018}.
We do not identify this matter-wave proposal with optical
interferometric motion readout.

The contribution consists of three results.
\begin{enumerate}
\item \textbf{Preparation-aware cancellation.}
A contrast cancelling \(d^2\) removes the normalized Gaussian
preparation-coherence signal, irrespective of whether the changed
coefficient arose from environment or gravity.

\item \textbf{Structured-force distinction.}
Support-displacement noise has a derived curvature-dependent
transfer. Its generator response is not proportional to persistent
gravity during a flat stage.

\item \textbf{Complete-record certificate.}
A four-time variation of the same staged architecture admits a
global bounded-statistic certificate based on the retained split
record, including finite sensing and measurement backaction.
\end{enumerate}

The four-time settings change the free-stage duration.
They retain the order and types of stages but are not the single
fixed design in the reference paper.
No additional post-split interrogation interval is introduced.

The central positive theorem is conditional on common physical
controls, a flat-stage model, calibrated competing diffusion, and
moment/error bounds.
It is not merely a local derivative-rank statement.
Nor is it evidence that the proposed certificate has useful
apparatus-specific power: its retained second-moment signal can be
much smaller than a coherence signal.

All mathematical prerequisites used below are stated and proved.
En01-A \cite{En01A} supplies conceptual context on accessible records,
not finite-dimensional premises for continuous-variable propagation.
Unrelated field-theory and manuscript-registry constructions are not
assumptions of this paper.

\section{Model boundary and common physical controls}
\label{sec:model}

\subsection{Centre-of-mass dynamics}

The retained system is a fixed-orientation rigid body moving along
one coordinate \(x\), with conjugate momentum \(p\) and mass \(M\).
On a stage with curvature \(\kappa(t)\),
\[
H(t)=\frac{p^2}{2M}+\frac12\kappa(t)x^2.
\]
Positive, zero, and negative curvature correspond to harmonic, flat,
and inverted quadratic stages.

A common environmental localization term is
\[
-D_{\mathrm{known}}(t)[x,[x,\rho]].
\]
Cooling may instead have an admitted linear Gaussian drift and
diffusion model.
We allow its output to depend on the hypotheses, provided it is
the same preparation procedure across the four time settings of
each hypothesis.

\begin{assumption}[Admitted dynamics and instruments]
\label{ass:admitted}
All propagators and sensing instruments used in a comparison are
CPTP when their complete outcomes are retained.
The unbounded operators have admitted realizations on the required
moment domains.
The moments differentiated below exist and justify the displayed
adjoint-generator identities.
Continuous record spaces are standard Borel spaces.
Any instrument approximation is accompanied by an explicit
complete-law error or a separately derived moment-error bound.
\end{assumption}

This assumption does not assert that arbitrary unbounded formal
master equations define instruments.
It fixes the domain requirements of the theorems.

\subsection{The declared gravitational contribution}

For intrinsic mass measure \(\mu_{\mathrm m}\) of total mass \(M\),
Gaussian smearing of Cartesian standard deviation \(\rc>0\) gives
\[
\widehat f_{\rc}(k)
=e^{-\rc^2|k|^2/2}\widehat\mu_{\mathrm m}(k).
\]
We use the decoherence-only generator
\begin{equation}\label{eq:G}
\GG_{\rc}(\rho)
=\int_{\R^3}
\left(e^{ik_ex}\rho e^{-ik_ex}-\rho\right)Q_{\rc}(\dd k),
\qquad
Q_{\rc}(\dd k)=
\frac{G}{2\pi^2\hbar}
\frac{|\widehat f_{\rc}(k)|^2}{|k|^2}\dd^3k .
\end{equation}
Here \(k_e\) is the component along the retained motion.
This is the normalization corresponding to
\[
-\frac{G}{2\hbar}\int\!\!\int
\frac{[\widehat\mu_{\rc}(r),
[\widehat\mu_{\rc}(r'),\rho]]}{|r-r'|}\dd r\,\dd r'.
\]
The alternative adds \(\lambda\GG_{\rc}\), \(\lambda\ge0\).

Since \(|\widehat\mu_{\mathrm m}|\le M\),
\[
Q_{\rc}(\R^3)\le
\frac{GM^2}{\sqrt\pi\hbar\rc}.
\]
The kick generator is bounded on trace class, with induced norm
at most twice that rate.
Its exponential is the Poisson mixture of the normalized random
unitary kick channel.
Thus its decoherence dynamics is well defined independently of
the unbounded Hamiltonian; adding it to an admitted baseline is
the ordinary bounded perturbation problem.

Define
\begin{equation}\label{eq:A}
A_{\rc}=\frac12\int k_e^2Q_{\rc}(\dd k),
\qquad
\Gamma_{\rc}(d)=\int[1-\cos(k_ed)]Q_{\rc}(\dd k).
\end{equation}
All kick moments needed here are finite for \(\rc>0\).
The quadratic comparison is
\[
\GG_{\rc}^{(2)}=-A_{\rc}[x,[x,\cdot]].
\]

\begin{lemma}[Exact second-moment action]
\label{lem:momentsG}
The full regularized generator and its quadratic comparison have
the same action on \(x^2,(xp+px)/2,p^2\):
\[
\GG_{\rc}^*(x^2)=0,\qquad
\GG_{\rc}^*((xp+px)/2)=0,\qquad
\GG_{\rc}^*(p^2)=2\hbar^2A_{\rc}\id.
\]
\end{lemma}

\begin{proof}
Conjugation of \(p\) by a position kick shifts it by
\(\hbar k_e\), up to a sign convention irrelevant after integration.
Evenness of \(Q_{\rc}\) cancels the linear term.
The quadratic term integrates to \(2\hbar^2A_{\rc}\).
The position is unchanged.
The same commutator calculation proves the quadratic-model identities.
\end{proof}

The equality is about second moments, not equality of states.
Regularized kick evolution is generally non-Gaussian.

\subsection{Fixed units do not mean hypothesis-dependent apparatus}

A physical split filter below contains the coefficient
\[
\eta=\chi/\sigma^2,
\]
with units of inverse length squared.
Changing the bookkeeping length \(\sigma\) requires
\(\chi\mapsto\eta\sigma^2\) to keep the same apparatus.
Holding \(\chi\) fixed while setting
\(\sigma=\xi_h\) separately for the hypotheses changes the physical
measurement.

The same principle applies to sensor strength, curvature, duration,
acceptance windows, phase compensation, and electronic response.
They are setting specifications, not quantities that may be silently
redefined using the unknown hypothesis.

\section{Physical support-displacement noise}
\label{sec:force}

\subsection{White-noise convention and transfer}

Let \(u(t)\) be a centred classical support displacement.
For a quadratic potential,
\[
\frac12\kappa(t)(x-u(t))^2
=
\frac12\kappa(t)x^2-\kappa(t)u(t)x
+\frac12\kappa(t)u(t)^2.
\]
The last term is scalar and contributes no commutator.
For an exactly quadratic translated potential this expansion is exact.
For a general potential, its neglected terms require a local
approximation bound.

We use the two-sided angular-frequency convention
\[
C_u(\tau)=\mathbb E[u(t+\tau)u(t)],
\qquad
S_u(\omega)=\int_{\R}e^{i\omega\tau}C_u(\tau)\dd\tau,
\]
\[
C_u(\tau)=\frac1{2\pi}\int_{\R}
e^{-i\omega\tau}S_u(\omega)\dd\omega.
\]
The white convention is
\[
C_u(\tau)=S_u\delta(\tau),
\]
so the two-sided spectrum is the constant \(S_u\).
One-sided spectra require their corresponding conversion; no
one-sided factor is inserted silently.

\begin{proposition}[White support-noise generator]
\label{prop:force}
In the white Gaussian displacement model, averaging the stochastic
Hamiltonian gives
\begin{equation}\label{eq:vib}
\LL_{\mathrm{vib}}(t)\rho
=
-\frac{\kappa(t)^2S_u}{2\hbar^2}[x,[x,\rho]].
\end{equation}
For \(\kappa=\pm M\omega^2\), its coefficient scales as
\(M^2\omega^4S_u/(2\hbar^2)\).
It is zero when \(\kappa=0\).
\end{proposition}

\begin{proof}
Represent the force term by
\(-\kappa x\sqrt{S_u}\circ\dd W_t\) in the stochastic Hamiltonian.
The Stratonovich density equation has stochastic superoperator
\[
B\rho=\frac{i\kappa\sqrt{S_u}}{\hbar}[x,\rho].
\]
Its It\^o correction is
\(\frac12B^2\rho
=-\kappa^2S_u[x,[x,\rho]]/(2\hbar^2)\).
The stochastic integral has zero mean.
The moment-domain or random-unitary realization in
Assumption~\ref{ass:admitted} justifies the averaging.
\end{proof}

This transfer agrees with the curvature scaling discussed for
harmonic vibration in the reference proposal \cite{Pino2018}.
A resonant colored-noise formula is not a substitute for
\eqref{eq:vib} in every stage.

\begin{proposition}[Failure of scalar generator absorption]
\label{prop:noabsorb}
Suppose the only uncertain force source is the white support amplitude
\(S_u\), shared across stages.
Suppose gravity contributes
\(-\lambda A_{\rc}[x,[x,\cdot]]\) with
\(\lambda A_{\rc}>0\) throughout a nonzero flat interval.
Changing \(S_u\) cannot match the two total generators on that interval,
and hence cannot match them across the complete protocol by the
scalar absorption construction of En01-B.
\end{proposition}

\begin{proof}
On the flat interval, the support-noise coefficient is zero for every
value of \(S_u\), while the gravity coefficient is positive.
The double commutator is a nonzero superoperator on its admitted
domain.
Thus the generators cannot coincide there.
\end{proof}

Failure of this matching construction does not exclude accidental
agreement after a restricted detector.
The constructive theorem below supplies a separate observation-level
argument.

\subsection{Colored noise and stage memory}

For deterministic quadratic dynamics let \(T(t,u)\) be its
two-dimensional phase-space transition matrix, and let
\(\ep=(0,1)^\top\).
The noise-induced displacement of \(R=(x,p)^\top\) is
\[
\delta R(t)=
\int_0^t T(t,u)\ep\,\kappa(u)u(u)\dd u,
\]
where the repeated letter \(u\) in the argument of the stochastic
process denotes time, not a second independent noise.
Equivalently, writing the time variables as \(a,b\), its covariance is
\begin{equation}\label{eq:colored}
N_u(t)=
\int_0^t\!\!\int_0^t
T(t,a)\ep\,\kappa(a)
C_u(a,b)\kappa(b)\ep^\top T(t,b)^\top
\dd a\,\dd b.
\end{equation}

\begin{proposition}[Colored transfer]
\label{prop:colored}
Equation~\eqref{eq:colored} is the exact added covariance for classical
Gaussian support noise independent of the initial quantum state,
under the stated quadratic dynamics.
It includes correlations between different curved stages.
The white limit reduces to
\[
N_u(t)=S_u\int_0^t
\kappa(a)^2T(t,a)\ep\ep^\top T(t,a)^\top\dd a.
\]
\end{proposition}

\begin{proof}
The force-driven linear Hamilton equations give the stated
\(\delta R\).
Its mean is zero.
Take the expectation of
\(\delta R\delta R^\top\) and use the displacement covariance.
Independence removes initial-state/noise cross terms.
The delta correlation reduces the double integral to the single one.
\end{proof}

No force from this mechanism acts during an exactly flat interval.
Nevertheless, colored noise can correlate forces before and after
that interval.
The finite-sensing theorem below uses independent white increments;
a colored extension must carry these cross-stage terms or bound their
departure from the white model.

\section{Gaussian preparation and outcome-conditioned splitting}
\label{sec:preparation}

\subsection{Covariance propagation}

For a centred Gaussian state write
\[
V=\begin{pmatrix}v_x&c\\c&v_p\end{pmatrix},
\qquad
c=\tfrac12\langle xp+px\rangle,
\qquad
\det V\ge\hbar^2/4.
\]
For quadratic Hamiltonian and localization coefficient \(D(t)\),
\begin{equation}\label{eq:covariance}
\dot V=B(t)V+VB(t)^\top+2\hbar^2D(t)\ep\ep^\top,
\qquad
B(t)=
\begin{pmatrix}0&1/M\\-\kappa(t)&0\end{pmatrix}.
\end{equation}
Hence
\begin{equation}\label{eq:covsolution}
V(t)=T(t,0)V(0)T(t,0)^\top+
2\hbar^2\int_0^t
D(a)T(t,a)\ep\ep^\top T(t,a)^\top\dd a.
\end{equation}

These equations follow by moving the Hamiltonian and double
commutators to the three quadratic observables.
They preserve Gaussianity in the quadratic localization model.
They also hold for second moments of non-Gaussian states under the
regularized kick model, by Lemma~\ref{lem:momentsG}.

For an inverted stage with curvature \(-M\omega^2\),
\[
T(t,0)=
\begin{pmatrix}
\cosh(\omega t)&\sinh(\omega t)/(M\omega)\\
M\omega\sinh(\omega t)&\cosh(\omega t)
\end{pmatrix}.
\]
The curvature contains \(M\), not \(M^2\).
The latter would have the wrong dimensions in the Hamiltonian.

For a flat interval of duration \(t\), constant coefficient \(D\)
gives
\begin{equation}\label{eq:free}
\begin{split}
v_x(t)&=v_{x,B}+\frac{2c_B}{M}t+\frac{v_{p,B}}{M^2}t^2
+\frac{2\hbar^2D}{3M^2}t^3,\\
c(t)&=c_B+\frac{v_{p,B}}M t+\frac{\hbar^2D}{M}t^2,\\
v_p(t)&=v_{p,B}+2\hbar^2Dt .
\end{split}
\end{equation}

Cooling can be represented by its own linear Gaussian covariance
equation with drift \(B_{\mathrm c}\) and diffusion \(N_{\mathrm c}\).
It is not reproduced by substituting a thermal output state into a
closed-system harmonic stage.
For the restricted reference diagnostic we use the declared
post-cooling thermal covariance
\[
v_x(0)=\frac{\hbar(2\bar n+1)}{2M\omega_1},
\quad
v_p(0)=\frac{\hbar M\omega_1(2\bar n+1)}2,
\quad c(0)=0.
\]
This is an input conditional on the preparation characterization,
not a new cooling-performance claim.

\subsection{Gaussian kernel and point-pair coherence}

Fourier inversion of the Gaussian Wigner function gives
\begin{equation}\label{eq:kernel}
\rho(x,y)=\frac1{\sqrt{2\pi v_x}}
\exp\!\left[
-\frac{(x+y)^2}{8v_x}
-\frac{\det V}{2\hbar^2v_x}(x-y)^2
+\frac{ic}{2\hbar v_x}(x^2-y^2)
\right].
\end{equation}
Define
\[
\mathcal P=\frac{\hbar}{2\sqrt{\det V}},
\qquad
\xi=\mathcal P\sqrt{8v_x}.
\]
These are purity and the reference Gaussian coherence length,
respectively \cite{Pino2018}.

\begin{theorem}[Result A: preparation-aware cancellation]
\label{thm:prep}
For the kernel \eqref{eq:kernel},
\[
\mathcal V_{\mathrm{pair}}(d)
=
\frac{|\rho(d/2,-d/2)|}
{\sqrt{\rho(d/2,d/2)\rho(-d/2,-d/2)}}
=e^{-a_{\mathrm{pre}}d^2},
\]
where
\begin{equation}\label{eq:apre}
a_{\mathrm{pre}}
=
\frac1{\xi^2}-\frac1{8v_x}
=
\frac{4\det V-\hbar^2}{8\hbar^2v_x}\ge0.
\end{equation}
Consequently, every weight vector satisfying
\(\sum_iw_id_i^2=0\) obeys
\[
\sum_iw_i[-\log\mathcal V_{\mathrm{pair},h}(d_i)]=0
\]
for each hypothesis \(h\).

In particular, \(w=(5,-8,3)\) at
\(d_i=R(1/4,1/2,3/4)\) cancels all hypothesis changes encoded only in
\(a_{\mathrm{pre},h}\), including gravitational changes.
It does not cancel a separately justified nonquadratic
post-split response merely because that response is gravitational.
\end{theorem}

\begin{proof}
The off-diagonal numerator is
\((2\pi v_x)^{-1/2}e^{-d^2/\xi^2}\).
Both diagonal entries are
\((2\pi v_x)^{-1/2}e^{-d^2/(8v_x)}\).
Their ratio gives \eqref{eq:apre}.
The uncertainty relation makes its coefficient nonnegative.
The weighted logarithm is
\(a_{\mathrm{pre},h}\sum_iw_id_i^2\).
\end{proof}

For a pure Gaussian, \(\mathcal V_{\mathrm{pair}}=1\).
This does not imply high-probability preparation at every separation,
nor resolved fringes after a real detector.

A decomposition
\(Y=Y_{\mathrm{pre}}+Y_{\mathrm{post}}+Y_{\mathrm{readout}}\)
requires multiplicative visibility factorization or a controlled
approximation.
It is not a general identity for successive quantum instruments.

\subsection{One physical split instrument}

Let \(\eta>0\) be fixed in physical coordinates and let
\[
m_z(x)=\left(\frac2\pi\right)^{1/4}
e^{-(\eta x^2-z)^2},
\qquad
M_z=m_z(x)e^{i\phi(x)}.
\]
Then
\[
\int_{\R}M_z^\dagger M_z\dd z=\id.
\]
For hypothesis \(h\),
\[
\tau_h(z)=M_z\rho_h^{\mathrm{pre}}M_z^\dagger,
\qquad
p_h(z)=\tr\tau_h(z),
\qquad
s_h(W)=\int_Wp_h(z)\dd z .
\]
For a Gaussian position marginal,
\begin{equation}\label{eq:splitlaw}
p_h(z)=
\sqrt{\frac2\pi}\int_{\R}
e^{-2(\eta x^2-z)^2}
\frac{e^{-x^2/(2v_{x,h})}}{\sqrt{2\pi v_{x,h}}}\dd x.
\end{equation}
Thus the outcome law depends on position variance, not directly on
purity or point-pair coherence.

The normalized likelihood is the classical representation
\[
Z=\eta X^2+N,\qquad N\sim N(0,1/4),
\]
with \(X\) distributed according to the input position density.
It is the physical outcome law, not a claim that physical outcomes are
the reference Wiener innovations.

\begin{proposition}[Point-pair filtering identity]
\label{prop:filter}
For any continuous positive kernel and a diagonal filter \(g_z(x)\),
the normalized point-pair coherence magnitude is unchanged wherever
\(g_z(x)g_z(y)\ne0\) and both input diagonal entries are positive:
\[
\frac{|\tau_z(x,y)|}{\sqrt{\tau_z(x,x)\tau_z(y,y)}}
=
\frac{|\rho(x,y)|}{\sqrt{\rho(x,x)\rho(y,y)}}.
\]
The identity is unaffected by normalization of the selected state.
\end{proposition}

\begin{proof}
The numerator acquires \(|g_z(x)g_z(y)|\), and the denominator acquires
the same factor.
A common trace normalization also cancels.
\end{proof}

This is a pointwise identity.
It is not an assertion that diagonal filtering preserves every
two-mode visibility or every measured fringe contrast.

For the diagonal density in \(\tau_h(z)\), nonzero maxima exist when
\(z>1/(8\eta v_{x,h})\).
Their separation is
\begin{equation}\label{eq:peaks}
d_h(z)^2=\frac{4z}{\eta}-\frac1{2\eta^2v_{x,h}}.
\end{equation}
This follows by differentiating
\(-2(\eta x^2-z)^2-x^2/(2v_{x,h})\).
The inverse square root of the negative log-density curvature at a
peak is
\[
\ell_{\mathrm{loc},h}(z)=\frac1{2\eta d_h(z)}.
\]
This explicitly defined local RMS-width approximation is not assigned
an ambiguous ``packet width'' convention.

Even at the same apparatus outcome \(z\), the conditional peak
locations can differ between hypotheses.
A window selected through a hypothesis-dependent \(d_h(z)\) is not
one common acceptance rule.
Use fixed \(z\)-windows or calibrated hypothesis-independent bounds.

\section{Finite packet and finite-window corrections}
\label{sec:packets}

\subsection{A physical two-mode diagnostic}

Choose fixed normalized packet modes \(u_+,u_-\) with disjoint supports
within distance \(\ell\) of centres \(d/2,-d/2\).
Their real envelopes are nonnegative after one declared common phase
compensation.
For an unnormalized block \(\tau\), define
\[
P_\pm=\langle u_\pm,\tau u_\pm\rangle,
\qquad
B=\langle u_+,\tau u_-\rangle,
\qquad
\mathcal V_{\mathrm{mode}}=\frac{|B|}{\sqrt{P_+P_-}}.
\]
Positivity gives \(0\le\mathcal V_{\mathrm{mode}}\le1\).
This is a projected two-mode coherence, still not automatically the
final fringe visibility.

\begin{proposition}[Finite-width Gaussian bound]
\label{prop:width}
Suppose the filtered Gaussian kernel on these supports has the form
\[
\tau(x,y)=
\sqrt{n(x)n(y)}
e^{-a(x-y)^2}e^{i(\varphi(x)-\varphi(y))},
\qquad a\ge0,
\]
where, after the common compensation,
\[
|\varphi(x)-\varphi(y)|\le\theta<\pi/2
\]
on the union of the packet supports.
Assume \(P_\pm>0\).
Then
\[
\left|-\log\mathcal V_{\mathrm{mode}}-ad^2\right|
\le
4ad\ell+4a\ell^2-\log(\cos\theta).
\]
\end{proposition}

\begin{proof}
Put
\(A_\pm=\int u_\pm(x)\sqrt{n(x)}\dd x>0\).
Within either support, \(|x-y|\le2\ell\).
Across the supports,
\[
\max(d-2\ell,0)\le|x-y|\le d+2\ell.
\]
Taking real parts for a lower bound and absolute values for an upper
bound yields
\[
\cos\theta\,e^{-4a\ell^2}A_\pm^2
\le P_\pm\le A_\pm^2,
\]
\[
\cos\theta\,e^{-a(d+2\ell)^2}A_+A_-
\le |B|
\le e^{-a\max(d-2\ell,0)^2}A_+A_-.
\]
Divide by the bounds on \(\sqrt{P_+P_-}\) and take logarithms.
For \(d\ge2\ell\) the lower logarithmic endpoint differs from \(ad^2\)
by at most \(4ad\ell-\log(\cos\theta)\).
For \(d<2\ell\), the displayed larger bound remains valid.
The upper endpoint contributes \(4ad\ell+4a\ell^2\).
\end{proof}

The phase condition must hold for the same physical compensation under
both hypotheses.
Choosing a separate exact chirp cancellation for each unknown state
would not produce a common measurement.
The log bound is useful only if the resulting coherence remains
strictly positive.

\subsection{Outcome mixing can destroy coherence}

For \(z\in W\), project the corresponding block into the same two
output modes, possibly after a fixed record-conditioned routing map.
Write
\[
P_\pm(z)=\langle u_\pm,\tau(z)u_\pm\rangle,
\qquad
B(z)=\sqrt{P_+(z)P_-(z)}\,V(z)e^{i\alpha(z)}.
\]
Define
\[
H_W=
\frac{\int_W\sqrt{P_+(z)P_-(z)}\dd z}
{\sqrt{\int_WP_+(z)\dd z\int_WP_-(z)\dd z}}.
\]
Cauchy--Schwarz gives \(0\le H_W\le1\).

\begin{proposition}[Window integration certificate]
\label{prop:window}
If \(V(z)\in[V_-,V_+]\), \(V_->0\), and all \(\alpha(z)\) lie within
\(\theta_W<\pi/2\) of one common phase, then the coherence after
forgetting \(z\) satisfies
\[
H_W\cos\theta_W\,V_-
\le V_W\le H_WV_+.
\]
For \(H_W>0\),
\[
-\log V_W\in
[-\log V_+-\log H_W,\,
 -\log V_--\log H_W-\log\cos\theta_W].
\]
If parity gives \(P_+(z)=P_-(z)\) almost everywhere, then \(H_W=1\).
\end{proposition}

\begin{proof}
Integrate the positive amplitude
\(\sqrt{P_+P_-}V\).
Its projection along the common phase is at least the same integral
times \(\cos\theta_W\); its modulus is at most the integral.
Use the visibility bounds and divide by the integrated populations.
Parity makes the numerator defining \(H_W\) equal either population.
\end{proof}

Without the phase bound, two equal-amplitude outcome blocks with phases
zero and \(\pi\) can cancel completely.
Keeping \(z\) avoids this particular loss of phase information.

For a Gaussian preparation and a window whose peak separation lies
within \(\delta_d\) of \(\bar d\),
\[
|a d(z)^2-a\bar d^2|
\le a(2|\bar d|\delta_d+\delta_d^2).
\]
Combining this bound, Proposition~\ref{prop:width}, and
Proposition~\ref{prop:window} supplies a preparation residual for a
nominal attenuation contrast.
It does not justify replacing that preparation by one free common
offset.

\section{Finite squared-position sensing and complete records}
\label{sec:sensing}

\subsection{A declared eliminated-cavity sensing model}

During either sensing stage use the continuous measurement model
\begin{equation}\label{eq:SME}
\begin{split}
\dd\rho_{\mathrm c}={}&
\LL_{\mathrm{dyn}}\rho_{\mathrm c}\dd t
-\mu[x^2,[x^2,\rho_{\mathrm c}]]\dd t\\
&+\sqrt{2\epsilon_{\mathrm m}\mu}
\left(x^2\rho_{\mathrm c}+\rho_{\mathrm c}x^2
-2\tr(x^2\rho_{\mathrm c})\rho_{\mathrm c}\right)\dd W_t ,
\end{split}
\end{equation}
with record
\begin{equation}\label{eq:recordY}
\dd Y_t=
2\sqrt{2\epsilon_{\mathrm m}\mu}
\tr(x^2\rho_{\mathrm c})\dd t+\dd W_t.
\end{equation}
Here \(\mu\) has units \(\mathrm m^{-4}\mathrm s^{-1}\), and
\(0\le\epsilon_{\mathrm m}\le1\) is monitoring efficiency.
The dynamics includes the common mean sensor curvature, force noise,
and gravity during sensing.

Equation~\eqref{eq:SME} is a specified reduced sensing model,
consistent in form with continuous squared-position monitoring.
It is not a proof of cavity elimination for every apparatus.
Assumption~\ref{ass:admitted} requires its instrument realization.
The reference uses additional approximations to obtain its split
filter \cite{Pino2018}; we do not use those approximations to declare
finite sensing impulsive.

For a duration \(\tau\), define the recorded number
\[
Z=\frac{Y_\tau-Y_0}{2\sqrt\tau},
\qquad
\eta_\tau=\sqrt{2\epsilon_{\mathrm m}\mu\tau}.
\]
In the frozen-position efficient limit this gives
\(Z=\eta_\tau X^2+N\), \(N\sim N(0,1/4)\), and the earlier filter.
For finite motion, it has the exact mean
\begin{equation}\label{eq:finiteMean}
\mathbb E Z=
\frac{\eta_\tau}{\tau}
\int_0^\tau\langle x^2\rangle_t\dd t .
\end{equation}
The expectation is unconditional, so it uses the complete sensing
instrument, not an accepted normalized branch.

\subsection{Exact moment transfer during sensing}

With constant sensor curvature \(\kappa_{\mathrm s}\) and strength
\(\mu\), let \(m=(v_x,c,v_p)^\top\).
For either quadratic localization or full regularized kicks,
\begin{equation}\label{eq:sensorMoment}
\dot m=B_{\mathrm s}m+j_h(t),
\quad
B_{\mathrm s}=
\begin{pmatrix}
0&2/M&0\\
-\kappa_{\mathrm s}&0&1/M\\
8\mu\hbar^2&-2\kappa_{\mathrm s}&0
\end{pmatrix},
\quad
j_h(t)=
\begin{pmatrix}0\\0\\2\hbar^2D_{\mathrm s,h}(t)\end{pmatrix}.
\end{equation}
The measurement term contributes
\[
-\mu[x^2,[x^2,p^2]]=8\mu\hbar^2x^2.
\]
It contributes zero to the other two moment equations.

Define the sensor row
\begin{equation}\label{eq:r}
r^\top=
\frac{\eta_\tau}{\tau}
\int_0^\tau e_x^\top e^{B_{\mathrm s}u}\dd u,
\qquad e_x=(1,0,0)^\top .
\end{equation}
Then
\begin{equation}\label{eq:recordlinear}
\mathbb E_hZ=r_xv_{x,\mathrm{in}}
+r_cc_{\mathrm{in}}+r_pv_{p,\mathrm{in}}+b_{\mathrm s,h},
\end{equation}
where \(b_{\mathrm s,h}\) is the integrated source term.
It can depend on the hypothesis and support-noise amplitude, but is
common across settings with identical sensing specifications.

\begin{lemma}[Positive finite-sensing position transfer]
\label{lem:rx}
If \(\mu>0\), \(\epsilon_{\mathrm m}>0\), and \(\tau>0\), then
\(r_x>0\).
If \(\epsilon_{\mathrm m}=0\), then \(r=0\).
\end{lemma}

\begin{proof}
The unconditional double commutator with \(x^2\) is averaging over a
classical white stochastic quadratic Hamiltonian.
Its phase-space dynamics is a random linear symplectic evolution.
The homogeneous response of \(v_x(u)\) to an initial increment in
\(v_x\), with the other second moments unchanged, is
\(\mathbb E[T_{11}(u)^2]\ge0\).
This is also the \(11\)-entry of \(e^{B_{\mathrm s}u}\), by the same
closed moment equations.
It equals one at \(u=0\) and is continuous.
Its integral is strictly positive.
Multiplication by \(\eta_\tau/\tau>0\) proves the claim.
Zero monitoring efficiency makes \(\eta_\tau=0\).
\end{proof}

The quadratic-Hamiltonian random-unitary argument is an unconditional
moment construction.
It does not replace the conditioned measurement instrument.

\subsection{Complete protocol law}

Let \(\Phi_{1:3,h,i}\) denote preparation including the setting's
free-stage duration, \(\II_{4,h}(\dd z)\) the finite split instrument,
and \(\Phi_{5:6,h,z}\) later record-conditioned propagation.
Let \(\II_{7,h,z}(\dd y)\) be finite final sensing and let a common
electronic map produce a detector outcome \(b\).
The complete law includes
\[
\pi_{h,i}(\dd z,\dd y,b)
=
\tr\!\left[
\II_{7,h,z}(\dd y,b)
\Phi_{5:6,h,z}
\II_{4,h}(\dd z)
\Phi_{1:3,h,i}(\rho_{\mathrm{init}})
\right],
\]
together with cooling flags, rejected split records, and failures.

Accepted blocks are propagated unnormalized.
A rejected record is not removed from the attempted-run sample.
Later trace-preserving dynamics cannot change the already recorded
split marginal.
The constructive statistic below may therefore ignore final outcomes
without pretending that the final sensing instrument is common at
the input of its finite interaction.

\section{Structured-force identifiability from four free times}
\label{sec:resultB}

\subsection{Settings and declared competing sources}

Choose
\[
t_i=t_0+i\Delta,\quad i=0,1,2,3,
\qquad
t_0\ge0,\quad\Delta>0,
\]
and use weights
\[
a=(-1,3,-3,1).
\]
They satisfy
\[
\sum_i a_it_i^j=0\quad(j=0,1,2),
\qquad
\sum_i a_it_i^3=6\Delta^3.
\]

The cooling and boost procedure is identical across these settings.
Its output second moments may depend arbitrarily on the hypothesis,
support amplitude, and other shared nuisance parameters.
The free interval is exactly flat.
Forces from white support displacement consequently vanish there.

Other admitted flat-stage sources have a constant aggregate
second-moment coefficient \(D_{\mathrm f}\ge0\).
Under the two hypotheses,
\[
D_{\mathrm f,0}=D_{\mathrm f},
\qquad
D_{\mathrm f,1}=D_{\mathrm f}+\lambda A_{\rc}.
\]
This notation includes any ordinary kick source with the same constant
second-moment heating action, not only Gaussian force diffusion.

The four sensing stages have identical duration, strength, curvature,
and electronic normalization.
White increments after the free interval are independent of earlier
ones.
Their integrated source offset is common across the four settings of
each hypothesis.

\begin{theorem}[Result B: exact complete-record mean signature]
\label{thm:fourtime}
Under the preceding specifications and moment-domain assumptions,
\begin{equation}\label{eq:fourtime}
T_h:=\sum_{i=0}^3a_i\mathbb E_hZ_i
=
\mathcal R\left(D_{\mathrm f}+h\lambda A_{\rc}\right),
\qquad
\mathcal R=\frac{4\hbar^2\Delta^3}{M^2}r_x>0.
\end{equation}
This identity holds for the full regularized kick model, not just its
Gaussian quadratic approximation.

All shared preparation second moments cancel, including their
gravitational and support-noise changes.
The sensor-source offset also cancels.
The retained gravitational contribution is its diffusion during the
flat interval.
\end{theorem}

\begin{proof}
Insert \eqref{eq:free} into \eqref{eq:recordlinear}.
Every contribution from \(v_{x,B},c_B,v_{p,B}\) is a polynomial in
\(t_i\) of degree at most two.
The sensor-source offset is constant.
The \(r_cc(t_i)\) diffusion term is quadratic, and the
\(r_pv_p(t_i)\) diffusion term is linear.
Only the cubic position term survives:
\[
r_x\frac{2\hbar^2D_{\mathrm f,h}}{3M^2}
\sum_i a_it_i^3
=
\frac{4\hbar^2\Delta^3r_x}{M^2}D_{\mathrm f,h}.
\]
Lemma~\ref{lem:momentsG} proves that the full kick model has the same
second-moment evolution.
Lemma~\ref{lem:rx} proves positivity.
\end{proof}

The theorem is compatible with En01-B's time-only obstruction.
That obstruction assumed proportional environmental and gravitational
attenuation templates.
Here the selected nuisance has zero coupling in the flat stage, while
the added diffusion persists there.

\subsection{Global distinction, local derivatives, and its limit}

Suppose \(D_{\mathrm f}\in[d_-,d_+]\) and
\(\lambda A_{\rc}\ge A_{\min}^{\mathrm{add}}>0\) over the tested
alternative family.
For known \(\mathcal R\), the untruncated mean ranges have gap
\begin{equation}\label{eq:rawgap}
G_T=
\mathcal R\left(A_{\min}^{\mathrm{add}}-(d_+-d_-)\right).
\end{equation}
A positive value gives global mean separation, not just local rank.

If sensor or timing calibration makes \(\mathcal R\) uncertain, use
the joint-family extrema
\[
G_T=
\inf_{\Theta_1}\mathcal R(\theta_1)
[D_{\mathrm f}(\theta_1)+\lambda A_{\rc}(\theta_1)]
-
\sup_{\Theta_0}\mathcal R(\theta_0)D_{\mathrm f}(\theta_0),
\]
with any certified mean departures subtracted.
Different free-time experiments cannot be assigned independent
arbitrary preparation covariances within these families.

For complete-record density or mass function \(p_i\), differentiation
of \eqref{eq:fourtime} yields, when differentiation under the moment
integral is justified,
\[
\sum_i a_i\int z\,\partial_{S_u}p_i(z)\dd z=0,
\qquad
\sum_i a_i\int z\,\partial_\lambda p_i(z)\dd z
=\mathcal R A_{\rc}>0.
\]
Thus gravity is transverse to the support-amplitude derivative at
this observation boundary when controls and competing diffusion are
fixed.
This calculation supplies a separating functional for those two
derivatives; it does not remove other nuisance columns automatically.

\begin{proposition}[The retained signal is total diffusion]
\label{prop:total}
If the null admits an additional free-stage momentum-diffusion
coefficient that can absorb \(\lambda A_{\rc}\), the four-time
second-moment signature is matched exactly.
In the quadratic gravity model, admitting this same shift throughout
all required stages can also match the entire dynamics.

Therefore the certificate identifies gravity only relative to
independently restricted competing sources.
It is not source identification from heating alone.
\end{proposition}

\begin{proof}
The first assertion follows from equality of
\(D_{\mathrm f}+h\lambda A_{\rc}\) in \eqref{eq:fourtime}.
For the second, matched quadratic coefficients give identical total
generators and therefore identical complete instrument laws.
\end{proof}

A small nonzero free curvature couples support displacement back into
that stage.
Its response must be calculated rather than called flat.
Likewise, nonconstant competing diffusion, setting-dependent reset
states, colored stage correlations, or a setting-dependent sensing
coefficient can create cubic or higher-order nuisance responses.
Their mean or complete-law departures enter the next theorem.

\section{A bounded complete-record statistical certificate}
\label{sec:statistics}

\subsection{Tail and implementation control}

The mean signature involves an unbounded record.
It cannot be inserted directly into Hoeffding's inequality.
For \(L>0\), let
\[
c_L(z)=\clip(z,-L,L).
\]
If \(\mathbb E_h Z_i^2\le B_{h,i}\), then
\begin{equation}\label{eq:clipping}
|\mathbb E_hZ_i-\mathbb E_hc_L(Z_i)|
\le\frac{B_{h,i}}L.
\end{equation}
Indeed,
\(|z-c_L(z)|\le z^2/L\).

Put
\[
B_h^{\mathrm w}=\sum_i|a_i|B_{h,i},
\qquad
B=B_0^{\mathrm w}+B_1^{\mathrm w}.
\]
These bounds must be uniform on the tested joint families.

A finite-sensor record bound is available directly from
\eqref{eq:recordY}.
Writing \(m_t=\tr(x^2\rho_{\mathrm c,t})\),
\[
Z=
\sqrt{\frac{2\epsilon_{\mathrm m}\mu}{\tau}}
\int_0^\tau m_t\dd t+\frac{W_\tau}{2\sqrt\tau}.
\]
Using \((a+b)^2\le2a^2+2b^2\), Jensen's inequality, and
\(m_t^2\le\tr(x^4\rho_{\mathrm c,t})\), we obtain
\begin{equation}\label{eq:Brecord}
\mathbb E Z^2
\le
4\epsilon_{\mathrm m}\mu
\int_0^\tau\langle x^4\rangle_t\dd t+\frac12.
\end{equation}
No independence between the innovation and the conditional state
was assumed in this estimate.

\begin{theorem}[Result C: complete-record gap and sample certificate]
\label{thm:certificate}
Randomize the four settings independently with probability \(1/4\).
Suppose their ideal mean ranges have gap \(G_T>0\), including any
certified mean-model errors.
Suppose the uniform record second-moment bounds above hold.

If the split record is always available, define the predeclared
bounded statistic
\begin{equation}\label{eq:statistic}
f(i,z,\text{other records})
=
\frac12+\frac{a_i}{6L}c_L(z).
\end{equation}
It belongs to \([0,1]\).
If instead the split record is independently erased with common
known probability \(1-\gamma\), \(0<\gamma\le1\), replace the second
term by \(a_i I_{\mathrm{observed}}c_L(z)/(6L)\), scoring an erased
record by \(1/2\).

Let actual complete attempted-run laws be within total variation
\(e_h\) of the corresponding specified laws, uniformly over the
joint families.
Then the actual oriented expectation gap is at least
\begin{equation}\label{eq:g}
g_*=
\frac{\gamma}{24L}\left(G_T-\frac{B}{L}\right)-e_0-e_1,
\end{equation}
where \(\gamma=1\) in the always-recorded case.

If \(g_*>0\), the midpoint of the enlarged null and alternative
expectation ranges gives a test with each error at most
\[
\exp(-Ng_*^2/2)
\]
after \(N\) independent attempted runs from a fixed actual family
member.
Thus
\[
N\ge\frac{2}{g_*^2}\log\frac1\alpha
\]
suffices for each error to be at most \(\alpha\).
A simultaneous calibration coverage probability at least \(1-\beta\)
gives unconditional error at most \(\alpha+\beta\).
\end{theorem}

\begin{proof}
Since \(\max_i|a_i|=3\) and \(|c_L|\le L\), the statistic is bounded.
Randomization gives
\[
\mathbb E_hf=
\frac12+\frac{\gamma}{24L}
\sum_i a_i\mathbb E_hc_L(Z_i).
\]
The independent-erasure condition gives the factor \(\gamma\).
Equation~\eqref{eq:clipping} reduces the ideal gap by at most \(B/L\).
For a statistic in \([0,1]\), total variation changes its expectation
by at most \(e_h\), giving \eqref{eq:g}.

The midpoint is at least \(g_*/2\) from each allowed expectation.
Hoeffding's one-sided inequality gives
\(\exp[-2N(g_*/2)^2]\) \cite{Hoeffding1963}.
Uniformity follows from the same expectation bounds for every family
member.
Finally, combine test and calibration failure events by a union bound.
\end{proof}

The independent-erasure model is a specific loss model.
State-dependent cooling failures, split-record loss, or selection
cannot be assigned the factor \(\gamma\) without validation.
They must be retained in the law and handled by a different response
model or error bound.
Split rejections after recording \(z\), and subsequent terminal
detector failures, do not erase this statistic's information.

\begin{corollary}[Required precision and optimal clipping scale]
\label{cor:optimal}
For fixed \(G_T>0\), \(B>0\), and \(\gamma>0\), the nominal gap
before complete-law errors is maximized at
\[
L_*=\frac{2B}{G_T},
\]
with value
\[
g_{\mathrm{nom,max}}=\frac{\gamma G_T^2}{96B}.
\]
Consequently a positive certificate of this form exists exactly when
\[
e_0+e_1<\frac{\gamma G_T^2}{96B},
\]
provided the clipping scale is unconstrained.
If hardware limits the record range, optimize only over admissible
\(L\).
\end{corollary}

\begin{proof}
Maximize
\(\gamma(G_T/L-B/L^2)/24\) over \(L>0\).
Its derivative vanishes at \(2B/G_T\), where the unique positive
maximum has the stated value.
\end{proof}

For known transfer and diffusion intervals,
\(G_T>0\) first requires
\(\lambda A_{\rc}>\Delta D_{\mathrm f}\) uniformly.
Even then the bounded record certificate can be extremely weak.
Large shared support noise cancels in the mean identity but increases
the moment bound \(B\), and hence increases the sample cost.

\subsection{When no positive bounded gap is supplied}

A valid alternative is a simultaneous confidence region for the
complete finite histogram.
For \(m\) bins and \(N\) independent attempted runs,
\[
|p_j-\widehat p_j|
\le
\sqrt{\frac{1}{2N}\log\frac{2m}{\alpha}}
\]
holds for all bins with probability at least \(1-\alpha\).
The bins include setting labels, preparation flags, split outcomes,
and terminal failures.

A model family is rejected only if no admissible joint nuisance value,
with its declared complete-law error, intersects this region.
This is a calibrated procedure even when no common linear statistic
has been found.
It is not an asserted apparatus-specific rejection.

\section{Restricted reference computation and error ledger}
\label{sec:computation}

\subsection{What is reproduced analytically}

Equations~\eqref{eq:covsolution} and \eqref{eq:free} give an independent
derivation of the quadratic Gaussian preparation evolution.
Equation~\eqref{eq:splitlaw} reproduces the normalized diagonal-filter
outcome calculation in physical coordinates.
These formula identities do not require numerical execution.

A useful restricted numerical comparison uses rounded design inputs:
\[
\begin{gathered}
R=1\,\mu\mathrm m,\qquad
\rho_{\mathrm{Nb}}=8570\,\mathrm{kg\,m^{-3}},\\
\omega_1/(2\pi)=2159.1\,\mathrm{Hz},\qquad
\bar n=0.057,\qquad
\omega_2/(2\pi)=50\,\mathrm{Hz},\\
t_2=17\,\mathrm{ms},\qquad
t_3=483\,\mathrm{ms},\qquad
S_u=10^{-36}\,\mathrm{m^2\,s}.
\end{gathered}
\]
For this diagnostic, thermal radiation is omitted and support noise
acts during the boost but not the exactly flat free stage.
Gravity is compared at the unsmeared continuous-sphere quadratic
coefficient \(A_0=GM^2/(2\hbar R^3)\).
This is not a microscopic unsmeared collapse prescription.

The reference's rounded comparison values are
\(\sqrt{v_x}\approx239\) nm at the split input,
\(\xi_0\approx607\) nm, and \(\xi_1\approx121\) nm
\cite[Appendix A.5]{Pino2018}.
The embedded artifact prints calculated values and discrepancies,
rather than inventing a completed comparison record.

The subtraction
\(v_xv_p-c^2\) is severely ill-conditioned after boosting.
The reference-formula diagnostic therefore uses 70-digit decimal
arithmetic, not a determinant formed by subtracting large
double-precision numbers.
Higher arithmetic precision is not a physical error bound.

\subsection{Three computational layers}

\begin{table}[ht]
\centering
\small
\begin{tabularx}{\linewidth}{@{}p{.25\linewidth}XX@{}}
\toprule
Layer & Completed in this article & Permitted conclusion\\
\midrule
Preparation &
Covariance and Gaussian-kernel identities;
rounded-input diagnostic supplied &
Analytical reference-model preparation formulas\\
Instruments and records &
Normalized filter; finite-sensor moment transfer;
complete-law construction &
Exact moment certificate in the declared reduced instrument model\\
Regularized full states &
Generator and moment identities;
error requirements identified &
No validated full-state continuum forecast\\
\bottomrule
\end{tabularx}
\caption{Agreement in a lower layer is not validation of a higher layer.
The cooling output used in the diagnostic is a declared input.}
\label{tab:layers}
\end{table}

\subsection{Fourth moments and propagation errors}

During quadratic Hamiltonian evolution, linear Gaussian diffusion,
and regularized kicks, polynomial moments through degree four form
a finite triangular system.
For efficient computation during sensing, use Weyl-ordered moments
\[
m_j^{(4)}=\langle x^{4-j}p^j\rangle_{\mathrm{Weyl}},
\qquad j=0,\ldots,4.
\]
The Hamiltonian contribution is
\[
\dot m_j^{(4)}
=
\frac{4-j}{M}m_{j+1}^{(4)}
-j\kappa m_{j-1}^{(4)}+\text{noise terms}.
\]
Quadratic monitoring contributes
\[
4\mu\hbar^2j(j-1)m_{j-2}^{(4)}.
\]
Gaussian diffusion contributes
\[
\hbar^2D\,j(j-1)
\langle x^{4-j}p^{j-2}\rangle_{\mathrm{Weyl}}.
\]
The regularized kick contribution has the same second-order term
with \(D=\lambda A_{\rc}\), plus, for \(j=4\),
\[
\lambda\hbar^4\int k_e^4Q_{\rc}(\dd k).
\]
Terms with out-of-range indices are omitted.
These identities follow by expanding a momentum shift and by the
exact quadratic Wigner commutator.
They permit a uniform fourth-moment enclosure once the initial
moments, parameter ranges, and integration error have been bounded.
They do not by themselves supply those numerical enclosures.

For full-state approximations, if an admitted exact propagator is
compared with an approximate Hermitian path \(\widehat\rho\) on a
common variation-of-constants domain, the residual
\[
r(t)=\dot{\widehat\rho}(t)-\LL(t)\widehat\rho(t)
\]
gives
\[
T(\rho(t),\widehat\rho(t))
\le
T(\rho(0),\widehat\rho(0))
+\frac12\int_0^t\|r(u)\|_1\dd u.
\]
This follows by propagating the initial difference and residual and
using CPTP contraction.
The same estimate applies to complete direct-sum instrument blocks.

\begin{table}[ht]
\centering
\small
\begin{tabularx}{\linewidth}{@{}p{.31\linewidth}X@{}}
\toprule
Application departure & Required accounting\\
\midrule
Cooling/reset uncertainty &
Shared setting law and calibrated moment region\\
Nonflat free potential &
Derived curvature/noise response and timing uncertainty\\
Colored support spectrum &
Cross-stage kernel, not a substituted resonant coefficient\\
Gas and magnetic forces &
Independent aggregate free-diffusion constraints\\
Finite cavity elimination &
Complete sensing-instrument or relevant moment error\\
Setting-dependent measurement strength &
Physical-coordinate transfer calibration\\
Split record saturation or loss &
Clipping model and recorded missing-outcome law\\
Finite packet and window averaging &
Width, phase, population imbalance, and tail bounds\\
Regularized continuum calculation &
Spatial/momentum tails, time error, kick quadrature, and residuals\\
Statistical calibration &
Joint coverage and independence or valid simultaneous construction\\
\bottomrule
\end{tabularx}
\caption{Inputs needed before the conditional certificate becomes an
apparatus-specific sensitivity claim.}
\label{tab:errors}
\end{table}

\section{Interpretation and conclusion}
\label{sec:conclusion}

The three-separation contrast is not an architecture-independent
gravitational witness.
For Gaussian preparation it cancels the normalized pre-split
coherence template of both hypotheses.
The mathematical nonquadratic contrast in En01-B remains valid for
its attenuation model, but that model is not automatically the
dominant signal-generation mechanism of the reference protocol.

A physically selected nuisance changes the analysis.
White support displacement couples through potential curvature.
It cannot imitate persistent flat-stage localization by changing one
shared amplitude.
The complete split record supplies a concrete way to exploit that
difference: four free times remove shared preparation second moments
and retain flat-stage diffusion, even with finite sensing and
quadratic measurement backaction.

The retained observable measures total diffusion.
Attribution to the selected gravity model still requires calibrated
bounds on ordinary flat-stage sources.
Its bounded-statistic margin also requires uniform moments,
implementation errors, and a valid loss model.
The results therefore provide both a warning about signal cancellation
and a conditional constructive alternative, rather than an unsupported
experimental exclusion.

No compiler run or numerical execution has been performed or claimed
in this source.
The mathematical results are complete as stated.
An apparatus-specific global histogram gap, validated continuum
propagation, cooling reproduction, and execution provenance remain
separate empirical and computational requirements.

\appendix

\section{Model interpretation and external constraints}
\label{app:interpretation}

\subsection{Rigid-body self-energy}

For fixed intrinsic density and orientation, translation-invariant
internal pair energy has the form
\[
U_{\mathrm{self}}=
-\frac G2\int\!\!\int
\mu_{\mathrm m}(\dd r)\mu_{\mathrm m}(\dd r')
W_{\rc}(|r-r'|),
\]
with the pair potential appropriate to the selected smearing.
A common centre-of-mass translation cancels inside \(r-r'\).
Thus \(U_{\mathrm{self}}\) is a scalar on the retained rigid-body
centre-of-mass Hilbert space and produces no centre-of-mass
commutator.
It does not remove the displaced-profile decoherence rate
\(\Gamma_{\rc}(d)\).

External masses, deformable internal states, rotation, or
centre-of-mass-dependent couplings change the model.
Their potentials must be specified separately.

Measurement-feedback semiclassical gravity can prescribe both a
regularized interaction and decoherence
\cite{TilloyDiosi2017}.
Testing only \eqref{eq:G} does not test the complete feedback theory.
This paper uses the decoherence-only interpretation.

\subsection{Mass specification and radiation}

A continuous homogeneous sphere, Gaussian-smeared microscopic
constituents, and a centre-of-mass wave packet are distinct inputs.
Packet width is not the regularization length.
Cross terms of microscopic constituent form factors cannot be
discarded by renaming the prescription.

The underground radiation study of Donadi and collaborators tests
the universal microscopic prescription considered there
\cite{Donadi2021}.
Transferring that result requires matched normalization, constituent
smearing, temporal spectrum, and charge-radiation dynamics.
It is not a universal exclusion of every macroscopic effective
decoherence generator.

\section{Naming, predecessor corrections, and debugging}
\label{app:debug}

All files generated here use the En01-C name.
The author-supplied En01-A file retains older Em02-A names internally;
that documentary inconsistency should be corrected in that file
without changing its mathematics.
This manuscript does not claim to have rewritten the supplied
predecessor bytes.

Two corrections to the supplied En01-B source are relevant:
\begin{enumerate}
\item For its three separations and weights, the moments are
\[
\sum_iw_i(d_i/R)^3=\frac{11}{32},
\qquad
\sum_iw_i(d_i/R)^5=\frac{239}{512}.
\]
The printed values \(15/32\) and \(435/1024\) in its rational
contrast proof are incorrect.
With the corrected moments,
\[
-\frac3{16}\frac{11}{32}
+\frac1{160}\frac{239}{512}
=-\frac{5041}{81920},
\]
so the stated final sphere contrast is unchanged.

\item The finite-sample log-inversion budget requires an exact
response law \(q=a+\kappa V\), or an explicit additional bound on
response misspecification.
A deterministic physical attenuation residual does not automatically
absorb a detector-response error before the logarithm.
Here the positive certificate uses the finite sensor record directly
and bounded complete-law errors, avoiding that identification.
\end{enumerate}

The following distinctions are enforced throughout:
\begin{enumerate}
\item Point-pair coherence, projected packet coherence, and measured
fringe visibility are not identified.
\item One common physical apparatus is used under both hypotheses.
\item Outcome windows and electronic rules are fixed before a run.
\item Accepted states are not substituted for unnormalized instrument
blocks in attempted-run likelihoods.
\item White and colored support noise use different transfer formulas.
\item Nonmatching generators are not automatically distinguishable
after a restricted detector.
\item The full regularized model need not preserve Gaussianity even
though its second-moment equations match the quadratic comparison.
\item Finite sensing includes measurement backaction.
\item The four-time variation is a declared design extension, not a
reproduction of the reference's one timing choice.
\item Mean cancellation does not imply a bounded-statistic gap without
uniform tails.
\item Cooling failures and state-dependent missing records are not
assigned an independent-erasure factor without justification.
\item All sample counts refer to attempted runs, not accepted fringes.
\end{enumerate}

\section{Artifact scope and execution provenance}
\label{app:artifact}

The generated program checks exact rational cancellation,
Gaussian pair coherence, common physical coordinates, covariance
propagation, finite-sensor moment transfer, and the four-time mean
identity.
It includes negative controls for changing physical units,
zero monitoring efficiency, and unrestricted competing free
diffusion.

Its restricted reference diagnostic evaluates preparation formulas
using rounded inputs and reports discrepancies against rounded
reference values.
It does not reproduce a cooling sensor, a finite split apparatus,
or a validated regularized continuum state.

A successful local invocation writes
\texttt{En01-C-run.json}, containing the script SHA-256, Python and
NumPy versions, UTC execution time, test status, and explicit scope
flags.
No successful invocation is asserted by this manuscript.

\begin{lstlisting}[language={}]
pdflatex En01-C.tex
pdflatex En01-C.tex
python En01-C-check.py
\end{lstlisting}

The generated script's build-computed MD5 identifier is:
\begin{center}
\texttt{\pdfmdfivesum file{En01-C-check.py}}
\end{center}
It identifies bytes and is not a proof or security certificate.

\subsection{Embedded regression program}

\lstinputlisting{En01-C-check.py}

\begin{thebibliography}{99}

\bibitem{Pino2018}
H.~Pino, J.~Prat-Camps, K.~Sinha, B.~P.~Venkatesh,
and O.~Romero-Isart,
\emph{On-chip quantum interference of a superconducting microsphere},
Quantum Science and Technology \textbf{3}, 025001 (2018).
doi:10.1088/2058-9565/aa9d15.
Reference version: arXiv:1603.01553v3.
Sections II--III and Appendix A.5.

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\end{thebibliography}

\end{document}